What is the highest power of 7 that divides 50!?

This question is from the topic Number Theory. It is a question on factorials and is about finding the highest power of a prime factor that will divide the factorial.

Question

What is the highest power of 7 that can divide 5000! without leaving a remainder? (5000! means factorial 5000)

1. 4998
2. 714
3. 832
4. 816

Correct Answer - 832. Choice (3)

Explanatory Answer

5000! = 5000 * 4999 * 4998 * 4997 * ............ 1.
Between 1 and 5000, there are numbers which will be divisible by 7, 7², 7³ and 7⁴.

There are = 714 numbers that are exactly divisible by 7 between 1 and 5000.

So there are 714 Sevens contained in these numbers.

There are = 102 numbers that are exactly divisible by 49 between 1 and 5000.

These numbers are also a part of the 714 numbers. But we add it again because they are divisible by 7² and hence to account for the second 7 in these numbers.

There are = 14 numbers that are divisible by 343 (i.e., 7³) between 1 and 5000.

These numbers will be a part of the previous set 714 and 102 - however, we add these 14 numbers to account for the third seven in these numbers as these numbers are multiples of 343 or 7³.

And finally, there are = 2 numbers that are divisible by 2401 (i.e. 7⁴).

Therefore, there will be a total of 714 + 102 + 14 + 2 = 832 sevens contained in 5000!
Hence the highest power of 7 that can divide 5000! without leaving a remainder is 832.

Correct answer choice (3)

Level of difficulty : Moderate to Difficult

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